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Indices (AQA GCSE Further Maths) : Revision Note

Dan Finlay

Written by: Dan Finlay

Reviewed by: Lucy Kirkham

Updated on

Laws of Indices

What are the laws of indices?

  • There are lots of very important laws (or rules)

  • It is important that you know and can apply these

  • Understanding the explanations will help you remember them

Law

Description

Why

a to the power of 1 equals a

anything to the power 1 is itself

6 to the power of 1 equals 6

a to the power of m cross times a to the power of n equals a to the power of m plus n end exponent

to multiply indices with the same base, add their powers

4 cubed cross times 4 squared equals open parentheses 4 cross times 4 cross times 4 close parentheses cross times open parentheses 4 cross times 4 close parentheses equals 4 to the power of 5

a to the power of m divided by a to the power of n equals a to the power of m over a to the power of n equals a to the power of m minus n end exponent

to divide indices with the same base, subtract their powers

7 to the power of 5 divided by 7 cubed equals fraction numerator 7 cross times 7 cross times up diagonal strike 7 cross times up diagonal strike 7 cross times up diagonal strike 7 over denominator up diagonal strike 7 cross times up diagonal strike 7 cross times up diagonal strike 7 end fraction equals 7 to the power of 2 space end exponent

open parentheses a to the power of m close parentheses to the power of n equals a to the power of m n end exponent

to raise indices to a new power, multiply their powers

open parentheses 14 cubed close parentheses squared equals open parentheses 14 cross times 14 cross times 14 close parentheses cross times open parentheses 14 cross times 14 cross times 14 close parentheses equals 14 to the power of 6

a to the power of 0 equals 1

anything to the power 0 is 1

8 to the power of 0 equals 8 to the power of 2 minus 2 end exponent equals 8 squared divided by 8 squared equals 8 squared over 8 squared equals 1

a to the power of negative n end exponent equals 1 over a to the power of n

a negative power is "1 over" the positive power

11 to the power of negative 3 end exponent equals 11 to the power of 0 minus 3 end exponent equals 11 to the power of 0 divided by 11 cubed equals 11 to the power of 0 over 11 cubed equals 1 over 11 cubed

a to the power of 1 over n end exponent equals n-th root of a

a power of an nth is an nth root

open parentheses 5 to the power of 1 half end exponent close parentheses squared equals 5 to the power of 1 half cross times 2 end exponent equals 5 to the power of 1 equals 5 so space space space 5 to the power of 1 half end exponent equals square root of 5

a to the power of m over n end exponent equals open parentheses n-th root of a close parentheses to the power of m equals n-th root of a to the power of m end root space

a fractional power of m over n means either

- do the the nth root first, then raise it to the power m

or

- raise it to the power m, then take the nth root

(depending on what's easier)

9 to the power of 3 over 2 end exponent equals 9 to the power of 1 half cross times 3 end exponent equals open parentheses 9 to the power of 1 half end exponent close parentheses cubed equals open parentheses square root of 9 close parentheses cubed or space space 9 to the power of 3 over 2 end exponent equals 9 to the power of 3 cross times 1 half end exponent equals open parentheses 9 cubed close parentheses to the power of 1 half end exponent equals square root of 9 cubed end root

open parentheses a over b close parentheses to the power of n equals a to the power of n over b to the power of n

a power outside a fraction applies to both the numerator and the denominator

open parentheses 5 over 6 close parentheses to the power of 3 space end exponent equals 5 over 6 cross times 5 over 6 cross times 5 over 6 equals 5 cubed over 6 cubed

open parentheses a over b close parentheses to the power of negative n end exponent equals open parentheses b over a close parentheses to the power of n equals b to the power of n over a to the power of n

flipping the fraction inside changes a negative power into a positive power

open parentheses 5 over 6 close parentheses to the power of negative 2 end exponent equals 1 over open parentheses 5 over 6 close parentheses squared equals 1 divided by open parentheses 5 over 6 close parentheses squared equals 1 divided by 5 squared over 6 squared equals 1 cross times 6 squared over 5 squared equals 6 squared over 5 squared equals open parentheses 6 over 5 close parentheses squared

Changing the base of a term

  • Sometimes expressions involve different base values

  • You can use index laws to change the base of a term to simplify an expression involving terms with different bases

    • For example 9 to the power of 4 equals left parenthesis 3 squared right parenthesis to the power of 4 equals 3 to the power of 2 cross times 4 end exponent equals 3 to the power of 8

    • Using the above can then help with problems like 9 to the power of 4 divided by 3 to the power of 7 equals 3 to the power of 8 divided by 3 to the power of 7 equals 3 to the power of 8 minus 7 end exponent equals 3 to the power of 1 equals space 3

Examiner Tips and Tricks

  • Index laws only work with terms that have the same base, so something like 23 × 52 cannot be simplified using index laws

Worked Example

Index laws example, A Level & AS Level Pure Maths Revision Notes

Solving Equations With Indices

How do I solve equations with indices using a change of base?

  • If two terms over the same base are equal, their their powers are equal

    • If a to the power of x equals a to the power of y then  x equals y

  • This is not true if there are other coefficients or terms in the way

    • e.g. 3 a to the power of x equals a to the power of y or a to the power of x equals a to the power of y minus 2

  • This is not true if the bases are different

    • e.g. a to the power of x equals b to the power of y ...

    • ...unless they can be changed into a common base

  • 4 to the power of x equals 8 to the power of y have bases 4 and 8 which can be written as 22 and 23 (over a common base of 2)

table row cell 4 to the power of x end cell equals cell 8 to the power of y end cell row cell open parentheses 2 squared close parentheses to the power of x end cell equals cell open parentheses 2 cubed close parentheses to the power of y end cell row cell 2 to the power of 2 x end exponent end cell equals cell 2 to the power of 3 y end exponent space space so space space 2 x space equals space 3 y end cell end table

  • Put brackets around your change-of-base to be able to use the index law open parentheses a to the power of n close parentheses to the power of m equals a to the power of n m end exponent

How do I solve equations with rational powers of x?

  • Use a to the power of m over n end exponent equals open parentheses n-th root of a close parentheses to the power of m equals n-th root of a to the power of m end root to convert a rational power into a root

    • If n-th root of x equals y then x equals y to the power of n

      • raise both sides to the power n

  • x to the power of 2 over 5 end exponent equals 4 becomes fifth root of x squared end root equals 4 which raises to the power 5 to give x squared equals 4 to the power of 5 equals 1024

    • solutions are x equals plus-or-minus square root of 1024 equals plus-or-minus 32

    • don't forget plus-or-minus rooting, plus-or-minus n-th root of blank, when n is even

  • Sometimes there is more than just x in a root

    • treat roots like brackets

    • cube root of x to the power of 5 plus 40 end root equals 2 cubes to x to the power of 5 plus 40 equals 2 cubed equals 8, which rearranges to x to the power of 5 equals negative 32

      • solution is x equals fifth root of negative 32 end root equals negative 2

      • you can root a negative if its an odd root (not even)

      • square root of a plus b end root not equal to square root of a plus square root of b so cube root of x to the power of 5 plus 40 end root not equal to cube root of x to the power of 5 end root plus cube root of 40

Examiner Tips and Tricks

  • In a non-calculator paper, choose which way to use a to the power of m over n end exponent equals open parentheses n-th root of a close parentheses to the power of m equals n-th root of a to the power of m end root to make calculations easier

    • x to the power of 3 over 4 end exponent equals 8 can be written fourth root of x cubed end root equals 8 or open parentheses fourth root of x close parentheses cubed equals 8 but the second way is easier to solve (the first way requires you to work out 84 then cube-root it!)

Worked Example

Solve the following equations

(a)table row cell blank to the power of blank end cell row blank end table  table row cell 8 to the power of x end cell equals cell 1 over 4 to the power of x cross times square root of 2 end cell end table.

Rewrite the right-hand-side of the equation using indices.

table attributes columnalign right center left columnspacing 0px end attributes row cell 8 to the power of x space end cell equals cell space 1 over 4 to the power of x space cross times space 2 to the power of 1 half end exponent end cell row blank equals cell space 4 to the power of negative x end exponent space cross times space 2 to the power of 1 half end exponent end cell end table

Change the bases so that all of the bases are the same using  2 squared space equals space 4 and  2 cubed space equals 8. 

open parentheses 2 cubed close parentheses to the power of x space end exponent equals space open parentheses 2 squared close parentheses to the power of negative x end exponent space cross times space 2 to the power of 1 half end exponent

Rewrite using the brackets law.

2 cubed to the power of x space end exponent equals space 2 to the power of negative 2 end exponent to the power of x space cross times space 2 to the power of 1 half end exponent

Rewrite the right-hand side using the law of indices: a to the power of x space cross times space a to the power of y space end exponent equals space a to the power of x space plus space y end exponent

.2 cubed to the power of x space end exponent equals space 2 to the power of open parentheses negative 2 x plus 1 half close parentheses end exponent

Now that the bases are the same with no extra components the powers can be equated and solved.

table row cell 3 x space end cell equals cell negative 2 x space plus space 1 half end cell row cell 5 x space end cell equals cell space 1 half end cell row cell 10 x space end cell equals cell space 1 end cell end table

bold italic x bold space bold equals bold space bold 1 over bold 10

(b)begin mathsize 20px style table row blank row blank end table end style3 x to the power of 2 over 3 end exponent minus 12 equals 0.

     

Isolate the x term by adding 12 and dividing by 3.

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 x to the power of 2 over 3 end exponent space end cell equals cell space 12 end cell row cell x to the power of 2 over 3 end exponent space end cell equals cell space 4 end cell row blank blank blank end table

Rewrite the left-hand side by converting the rational power into a root. 

open parentheses cube root of x close parentheses squared space equals space 4

Square root both sides, don't forget there could be a positive or a negative answer.

cube root of x space equals space plus-or-minus square root of 4 space equals space plus-or-minus 2

Cube both sides, remember to cube both the positive or a negative solutions.

x space equals space open parentheses plus-or-minus 2 close parentheses cubed space equals space plus-or-minus 8

bold italic x bold space bold equals bold space bold plus-or-minus bold 8 

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    Author
    Reviewer
    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

    Lucy Kirkham

    Reviewer: Lucy Kirkham

    Expertise: Head of STEM

    Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels.Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all.